Theorem 3.3 (Singular Inversion Theorem) 
Let \(C(z)\)  be a generating function such that \(C(0)=0\) , satisfying the equation \(C(z) = z \phi (C(z))\)  for \(\phi (z) = \sum _{n\ge 0} \phi _n z^n\)  a power series such that \(\phi _0 \ne 0\) , all \(\phi _n\)  are non-negative real numbers, and \(\phi (z) \ne \phi _0 + \phi _1 z\) .
Denote by \(R\)  the radius of convergence of \(\phi \)  at 0. Assume that \(\phi \)  is analytic at 0 (so that \(R>0\) ), that the characteristic equation \(\phi (z) - z \phi ^{\prime }(z) = 0\)  has a solution \(\tau \in (0,R)\)  (that is necessarily unique), and that \(\phi \)  is aperiodicAperiodicity is needed only for the third item below. The definition of aperiodicity is omitted from this paper, and can be found in [1]}. A sufficient condition for a power series to be aperiodic (which applies to all examples considered in this paper), is to have \(\phi _n >0\)  for all \(n\) .. Then the followings hold:
